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Chapter 3: Problem 15

Use synthetic division to perform the indicated division. Write the polynomial in the form \(p(x)=d(x) q(x)+r(x)\). \(\left(2 x^{3}+x^{2}+2 x+1\right) \div\left(x+\frac{1}{2}\right)\)

Short Answer

Expert verified
The division results in the polynomial \(2x^2 + 2\) with a remainder of 0.

Step by step solution

01

Set up the Synthetic Division

To perform synthetic division, we need the divisor in the form \(x - c\). Given \(x + \frac{1}{2}\), \(c = -\frac{1}{2}\). Write the coefficients of the dividend \(2x^3 + x^2 + 2x + 1\) in a row, which are \(2, 1, 2, 1\). We will use these coefficients for the procedure.
02

Perform the Division Process

Write the value of \(c = -\frac{1}{2}\) to the left and the first coefficient (2) below the line. Multiply \(-\frac{1}{2}\) by 2, place the result beneath the next coefficient (1), and add to get the new value \(1 + (-1) = 0\). Continue the process: multiply \(-\frac{1}{2}\) by 0 to get 0, add to 2 to get 2. Finally, multiply \(-\frac{1}{2}\) by 2 to get \(-1\), add to 1 to get 0.
03

Interpret the Results

The results from synthetic division are \(2, 0, 2\) as coefficients of the quotient polynomial, with the remaining term 0 as the remainder. This implies that the polynomial division results in a quotient \(2x^2 + 0x + 2\) with a remainder of 0.
04

Express in Polynomial Division Format

According to the polynomial division formula \(p(x) = d(x)q(x) + r(x)\), substitute back the values: \( (2x^3 + x^2 + 2x + 1) = (x + \frac{1}{2})(2x^2 + 2) + 0\). Thus, the polynomial division is complete with no remainder.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polynomial Division
Polynomial division is a process similar to the long division you might have learned with numbers, but this time, we deal with variables and their coefficients. In essence, it's a method to divide a polynomial by another polynomial of lower degree. This technique is particularly handy in simplifying complex polynomial expressions, either by simplifying them into more manageable pieces or by discovering factors.
  • It's used when you want to express a polynomial \( p(x) \) as a product of another polynomial \( d(x) \) and a quotient polynomial \( q(x) \), with some remainder \( r(x) \) possibly left over.
  • The expression follows this formula: \( p(x) = d(x)q(x) + r(x) \).
There are different methods of polynomial division, but **synthetic division** and **long division** in algebra are widely used. Synthetic division is streamlined for specific cases, making it a quick and effective tool.
Remainder Theorem
The Remainder Theorem is a useful concept that tells us what the remainder will be when dividing a polynomial \( p(x) \) by a linear divisor \( x - c \). The theorem states that the remainder of this division is simply the value \( p(c) \).
  • It simplifies the problem because you don't have to execute the entire division to find the remainder.
  • In the example provided, since the remainder is 0, \( x + \frac{1}{2} \) is actually a factor of \( 2x^3 + x^2 + 2x + 1 \).
  • This can be verified by evaluating the function at \( x = -\frac{1}{2} \) which should yield zero, reflecting the remainder theorem's essence.
Quotient Polynomial
After division, the quotient is simply the polynomial result of the division, without taking the remainder into account. This part of the division is vital since it represents what remains when the initial polynomial is divided by the divisor.
  • In synthetic division, the quotient polynomial is derived from the coefficients obtained through the process.
  • For the exercise example, the coefficients from the synthetic division gave us \( 2x^2 + 0x + 2 \).
  • This quotient shows how multiple times the divisor fits into the original polynomial.
Understanding the quotient polynomial is crucial because it provides insight into the structure and factors of the original polynomial expression.
Long Division in Algebra
Long division in algebra works similarly to long division with numbers. It's a step-by-step process used to divide larger polynomials by smaller polynomials. It requires careful organization and attention to detail, as each step builds on the last.
  • First, align the polynomials in descending order of degree.
  • Divide the leading term of the dividend by the leading term of the divisor, writing the answer in the quotient.
  • Multiply the entire divisor by this quotient term and subtract from the dividend.
  • Repeat these steps like a loop until all terms have been considered.
Although synthetic division is faster, understanding long division in algebra is equally important as it lays down the fundamental mechanics of division which can be applied even when synthetic division is not possible.

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Most popular questions from this chapter

In Exercises 31-40, you are given a polynomial and one of its zeros. Use the techniques in this section to find the rest of the real zeros and factor the polynomial. \(x^{3}-6 x^{2}+11 x-6, \quad c=1\)

Determine \(p(c)\) using the Remainder Theorem for the given polynomial functions and value of \(c\). If \(p(c)=0,\) factor \(p(x)=(x-c) q(x)\). \(p(x)=x^{4}+x^{3}-6 x^{2}-7 x-7, c=-\sqrt{7}\)

Use polynomial long division to perform the indicated division. Write the polynomial in the form \(p(x)=d(x) q(x)+r(x)\). \(\left(5 x^{4}-3 x^{3}+2 x^{2}-1\right) \div\left(x^{2}+4\right)\)

You are given a polynomial and one of its zeros. Use the techniques in this section to find the rest of the real zeros and factor the polynomial. \(x^{3}-24 x^{2}+192 x-512, \quad c=8\)

Find the real zeros of the polynomial using the techniques specified by your instructor. State the multiplicity of each real zero. \(f(x)=x^{3}+4 x^{2}-11 x+6\)
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